Chapter 6: Numerical solutions to boundary value problems

Governing equations in 3D:

Simplified (little) BVP in 1D:

that satisfy SD and SS relationships above for all x, and u(x) = uD on ϕ(AD)and

Find displacement field u(x), strain field ε(x), and stress field S(x)

Recall Big Picture:

Ch6-ApproxNumSols Page 1

The Ritz Method

Ch6-ApproxNumSols Page 2

Ch6-ApproxNumSols Page 3

Example: 1D Problem:

Ch6-ApproxNumSols Page 4

Matlab code for 1D Ritz method

Ch6-ApproxNumSols Page 5

Matlab code for 1D Ritz method (continued)

Ch6-ApproxNumSols Page 6

Results

Ch6-ApproxNumSols Page 7

Results

Ch6-ApproxNumSols Page 8

Summary of the method for obtaining approximate solutions:

Ch6-ApproxNumSols Page 9

Derivation of weak forms

Ch6-ApproxNumSols Page 10

Approximation functions

When choosing functions to approximate the real displacement u(x) and the virtual displacement ū(x),Make sure that they satisfy the continuity requirements imposed by the weak form (H1 for PVW) and,The approximation functions hi(x) are completei.e. they can converge to the exact solution in the limit i → ∞.

Condition number of K

In addition, one should try to make sure that the approximation functions are sufficiently different from one another. If the functions are not sufficiently different, it can lead to poorly conditioned system of equations K a = f

If condition number is large (~105 or larger) the computer will not be able to solve the system accurately.In order to keep the condition number small, we should use hi(x) functions that are linearly independent.

Orthogonal vectors and Orthogonal functions

Ch6-ApproxNumSols Page 11

Gram-Schmidt orthogonalization

In order to get smaller (better) condition numbers for the system matrix K, one should choose different (possibly orthogonal) approximating functions.It is possible to generate a set of orthogonal vectors (or functions) from a given set of linearly independent vectors (or functions) which are not necessarily orthogonal to each other.

Similarly for functions:

Ch6-ApproxNumSols Page 12

1D Finite Element Basis

Approximation (Alternative implementation of HEBC)

1D Finite Element Basis Functions

Ch6-ApproxNumSols Page 13

Ch6-ApproxNumSols Page 14

Similarly for the force vector

Ch6-ApproxNumSols Page 15

MATLAB Code

Ch6-ApproxNumSols Page 16

Element-wise computations for finite elements

Ch6-ApproxNumSols Page 17

Ch6-ApproxNumSols Page 18

Discretized weak form

Ch6-ApproxNumSols Page 19

Ritz and finite element methods for 2D and 3D problems

Ritz method Approximation:

Note: It may not always be easy find such functions for complicated shapes and boundary conditions.

Ch6-ApproxNumSols Page 20

Discretized weak form:

Approximations of strains and stresses

Finite Element approximations

In 2D In 3D

Ch6-ApproxNumSols Page 21

Finite Element "Element-wise" approximations 2D

Finite element "Element-wise" approximation 3D

Ch6-ApproxNumSols Page 22